What is a soft derivative? â It is an asset whose payoff is a function of some other variable, but that variable is not marketed. â Examples: options on profit, ...
Soft Derivatives David G. Luenberger October 2007
What is a soft derivative? z It
is an asset whose payoff is a function of some other variable, but that variable is not marketed. z Examples: options on profit, weather, or many other things.
Outline z Properties
and variations of CAPM z Properties and variations of zero-level pricing z Axiomatic pricing in continuous time z The abstract Black-Scholes equation z The extended Black-Scholes equation z Hedging
Linear Pricing z Consider
a set of n assets with payoffs (at the end of a year) y1, y2, y3, ... , yn and prices p1, p2, p3, ... , pn. z Linear pricing implies that a combination asset will have the combination price. That is, Σwi yi will have price Σwi pi.
CAPM 1⎡ cov(y, yM )(yM − p M R) ⎤ p= E(y) − 2 ⎢ ⎥⎦ R⎣ σM
2.2
2.2
Expected Return 2 yM
where yM is an efficient risky asset.
[
1 p = E (y ) − β y ,M (y M − p M R ) R
]
2
1.8
1.8
1.6
1.6
1.4
1.4
1.2
1.2
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
Sigma 0
0.2
0.4
(a) R f = 1.0
Expected Return
0.6
0.8
0
0
0.2
0.
(b) R f =1.
CAPM z It
is always true (if it exists) for the given set of assets.
Asset Outside Span y
M y1 y2
Price by Projection y
Set py = pm .
M m
y1 y2
Equivalence z The
price assigned by the CAPM (when it exists) is equal to the projection price (which always exists).
Hilbert Space for Pricing zH
is the set of all linear combinations of asset payoffs. z The inner product is (y1| y2) = E(y1 y2) = E(y1)E(y2) + cov (y1, y2) z Since H is finite dimensional, all subspaces are closed.
Minimum Norm Pricing Vector p=(g|y)/(g|g)
y
p=1 m0 g
M
Minimum-Norm Pricing 1⎡ cov(y, yM )(yM − p M R) ⎤ p= E(y) − 2 ⎢ ⎥⎦ R⎣ σM
2.2
2.2
Expected Return 2 yM
where now yM is the traded asset with price 1 and minimum norm.
[
1 p = E (y ) − β y ,M (y M − p M R ) R
]
2
1.8
1.8
1.6
1.6
1.4
1.4
1.2
1.2
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
Sigma 0
0.2
0.4
(a) R f = 1.0
Expected Return
0.6
0.8
0
0
0.2
0.
(b) R f =1.
Application to Pricing 2
Expected 1.8 Return
• y1 = R1 = 1.4, • σ1 = σ2 = .20 • Uncorrelated.
y2 = R2 = .8
1.6 1.4 1.2 1 0.8 0.6 0.4
•Implied risk-free return = 1.2 • p= E{y(-y1 + 2 y2}/.24 • yM = -y1 + 2 y2 • p = (R0)-1 [E(y) -cov(y, yM)(E(yM)- pM R0)/σM2] 0.2
0
0
0.2
Sigma 0.4
0.6
0.8
Add Risk-free Asset z Same
y1 and y2 as before. z Risk-free return is Rf z Critical value is Rf = 1.1 z In some cases there is no efficient risky portfolio and hence no CAPM formula. But there is always a minimum norm pricing formula.
2.2
2.2
Expected Return 2 yM
Expected Return
2.2
2
2
1.8
1.8
1.8
1.6
1.6
1.6
1.4
1.4
1.4
1.2
1.2
1.2
1
1
1
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0
Sigma 0
0.2
0.4
(a) R f = 1.0
0.6
0.8
0
Sigma 0
0.2
0.4
(b) R f =1.3
0.6
0.8
0
Expected Return
Sigma 0
0.2
0.4
(b) R f =1.1
0.6
0.8
The Dual Theorem y
M m0 θ
m'
Correlation Pricing Formula 1⎡ cov(y, yM )(yM − pM R)⎤ p = E(y) − 2 ⎥⎦ ⎢ R⎣ σM where now yM is an asset most correlated with y. (The magnitude and risk-free component are arbitrary.) Try it for y in M. Then yM = y is most correlated. We find p = R-1[ E(y) - E(y) + py R] = py.
Advantages z Solidly
based on projection theorem. z The correlation pricing formula for an asset y uses the prices of assets that are similar to y. k z It is a rigorous expression of “pricing by Stoc rt epo R comparables”.
Advantages z Solidly
based on projection theorem. z The correlation pricing formula for an asset y uses the prices of assets that are similar to y. z It is a rigorous expression of “pricing by comparables”.
Second Topic z Use
general portfolio theory to assign prices z Zero-level prices z Does result depend on utility function? z Applications
Coin Flip 3 H ? T 0
Add an asset z Suppose
asset with payoff x is not in the market but is available to you. What is the logical price? z Theorem. Assume there is no arbitrage in the original system. Then there is an open interval of the real line such that for px in this interval no arbitrage is possible. z For coin flip, what is the interval?
Zero-Level z Theorem.
There is a price px such that x is taken in the portfolio at zero level. Thus, the optimal portfolio does not change. z This is called the zero-level price. z For coin flip, what is the zero-level price?
Portfolio Problem z Maximize
E[U(y0)] z Sub to y0 = a1 y1 + a2 y2 +... +an yn + an+1Rf z W = a1 p1 + a2 p2 +... +an pn + an+1
Necessary conditions z E[U’(y0) yi]
= λ pi for each i and some
λ > 0. z Can find λ=RE[U’ (y0) ] z p=E[U’
(y0) y] / RE[U’ (y0) ]
Independent x z If
x is in dependent of all yi’s then we can find a unique zero-level price. z Since p=E[U’(y0)x]/Rf E[U’ (y0) ], it follows that px = E(x)/ Rf . Price is universal zero-level price. z There are other cases where the zerolevel price is universal.
Continuous-time Framework z Market
prices follow stochastic processes, such as dxi(t) = μi xi (t) dt + σi xi (t) dzi. z There is a risk free asset with rate of return r. z Frictionless trading is possible at every instant. z Everyone is a price taker.
General Idea of Operational Calculus z We
only need to know how at time t to price payoffs at time t + dt. z We use only knowledge of how to value risk free payoffs and marketed payoffs. z Only first-order terms in dt are relevant.
An Operational Calculus FOUR AXIOMS: z Pricing a constant: If C is constant, then P{C}=C·(1- r dt) z Pricing a marketed quantity: If x is an evolving market variable that neither pays dividends nor requires holding costs, P{x + dx} = x. z P{dt} = dt z P is linear.
Main Application z Since
x is a constant x = P{x + dx} = (1- r dt) x + P{dx} Hence rx dt = P{dx}
Fundamental Pricing Equation zA
value function on the span of marketed assets must satisfy rV(x,t) dt = P{dV(x, t)}
This is the general Black--Scholes Equation
Standard Black--Scholes Equation 1 2 dV = Vt dt + Vx dx + Vxx (dx) 2 1 2 2 = Vt dt + Vx dx + Vxxσ x dt 2 1 2 2 P{dV} = Vt dt + Vx rxdt + Vxxσ x dt 2 rV(x,t) dt = P{dV(x, t)}
1 2 2 rV = Vt + Vx rx + Vxxσ x 2
Extend outside the marketed space z Apply
similar idea to payoffs outside the marketed space. z Use instantaneous projection (as CAPM uses projection).
The Extended framework dxe = μe xe dt + σ e xe dze
Underlying process
dxi = μi xi dt + σ i xi dzi i = 1,2,...,n Stock processes F(xe (T ))
Payoff function
Instantaneous Projection z Definition
of price
p y = P{y | M } z Pricing equation V (xet) = P{V (xe ,t) + dV (xe ,t) | M } Price of y when projected onto M
rV(xe ,t)dt = P{dV (xe ,t) | M } z For standard case
General extended B-S equation
dV = [Vt + Vxe μ e xe + Vxe xe σ x ]dt + Vx e σ e xe dz e 1 2
2 2 e e
Projection of dze {dze | M } = adt + bdzm E [(dze − adt − bdz m )dt ] = 0 E [(dze − adt − bdz m )dzm ] = 0 a = 0, b = ρ em
{dze | M } = ρem dzm
Price of dze z From
operational calculus:
rxm dt = P {dxm } = P {μ m xm dt + σ m xm dzm } = μm xm dt + σ m xm P {dz m } z Hence (r − μ m ) P{dzm } = dt σm z Finally P{dze } =
(r − μ m )
σm
ρ emdt
Beta Form We have
ρ em P{dze | M } = (r − μ m )dt σm Define Then
2 βem = σ em / σ m
βem P{dze | M } = (r − μ m )dt σe
The Equation (Standard Version)
dxe = μe xe dt + σ e xe dze Underlying process dxi = μi xi dt + σ i xi dzi i = 1,2,..., n Stock processes F ( xe (T )) Payoff function
rV(xe ,t) = Vt (xe ,t) + Vxe (xe ,t)xe [ μe − βem (μm − r)] 1 2 2 + Vx e xe (xe ,t)xe σ e 2
V (xe ,T) = F(x e )
Main equation
Boundary condition
Alternate xm’s z Most-correlated –
Valid for xe and all its derivatives
z Dual – –
Asset
Valid for all assets. Always exists
z Markowitz – –
marketed asset.
portfolio
Valid for all assets May not exist
Universal Propery maxα E{U(W(T)} subject to dW(t) = α ′x dx + α e dV (xe , t) W (0) = W0 α ′x x + α eV (xe ,t) = W (t) Solution:
αe = 0
Universal Property
J(xe ,W,t) = U (W (T))
J(xe ,W,t) = maxα E{J(xe + dxe ,W + dW,t + dt)} subject to dW(t) = α ′x dx + α e dV (xe , t) W (0) = W0 α ′x x + α eV (xe ,t) = W (t) Solution:
αe = 0
Optimal Replication z Can
replicate at each instant so as get best fit. z Begin with V(xe, 0) dollars and at each instant allocate fraction γ to the mostcorrelated asset and 1- γ in the risk free asset, where
γ=
Vx e (xe ,t)xe βem V (xe ,t)
Extended Equation Properties
z z z z z z z z z
Prices are linear. Instantaneous projection consistent. No arbitrage opportunities generated. Gives universal zero-level price. Agrees with B--S if pricing a derivative. Gives best replication (or best hedge). Can be based on market or most-correlated portfolio. Easy to implement (similar to B--S). Gives formula for total error after hedge.
Extensions z Discrete
time (lattice) method z Risk-neutral form z More complex dynamics z Expectations of any variable z Hedging
Risk-Neutral Process dxe = μe xe dt + σ e xe dze
Underlying process
dxi = μi xi dt + σ i xi dzi i = 1,2,...,n Stock processes dxe = ω em xe dt + σ e xe dze
ω em = μe − βem (μ m − r)
Risk-Neutral Process
Example z Consider
an option on the variable xe(T) with strike price K. z xe may be estimated stock value of untraded company. z xe may be revenue or profit z Can get closed-form expression
Option formula 1 rV(xe ,t) = Vt + Vx e ωxe + Vxe x e σ e2 xe2 2 ω = μe − βem (μm − r)
V ( xe , t ) = xe e d1 =
(ω − r)( T − t)
N (d1 ) − Ke
− r (T − t )
1 2 ln(xe / K) + (ω + σ e )(T 2
d2 =
N ( d2 )
− t)
σe T −t 1 2 ln(xe / K) + (ω − σ e )(T − t) 2
σe T −t
Specific Example z Standard –
K = 60, S= 62, T = 5 mo, r = 10%, σ = 20% Value = 5.85
z Non – –
Option
traded version
K = 60, S = 62, T = 5 mo, r =10%, σe = 20%, μe = 8%, σm = 15%, μm = 14%, ρ=.7 Value = 4.79
Projection Error (Risk that cannot be hedged) z Let
S(xe, t) be the variance at T as seen at t. We may find S from 0 = E(dS) + δ2 z Or, in detail, 1 St + S xe μe xe + S xe x e σ e2 xe2 2
S(xe ,T) = 0
2 + [Vx e σ e xe ]2 (1 − ρ em )=0
Summary Conclusion z B-S
pricing renders a derivative redundant, in the sense it can be constructed from existing securities. z Extended pricing renders a soft derivative irrelevant, in the sense that no risk-averse investor will want it in a portfolio.
References (available on Stanford website) Luenberger, David G. (1999), Projection Pricing, Journal of Optimization Theory and Applications, vol. 109, no. 1, April 2001, 1-25. z Luenberger, David G. (2000), A Correlation Pricing Formula, Journal of Economic Dynamics and Control, 26, (2002), 1113-1126. z Luenberger, David G. (2001), Arbitrage and Universal Pricing, Journal of Economic Dynamics and Control, 26, (2002), 1613-1628. z Luenberger, David G. (2003), Pricing a Nontradeable Asset and Its Derivatives, Journal of Optimization Theory and Applications, Vol. 121, No. 3, June 2004, 465-487. z
